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Properties

Label 2-177-177.176-c5-0-2
Degree $2$
Conductor $177$
Sign $-0.941 + 0.336i$
Analytic cond. $28.3879$
Root an. cond. $5.32803$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.3·2-s + (−12.6 + 9.17i)3-s + 75.7·4-s + 9.35i·5-s + (130. − 95.2i)6-s + 67.1·7-s − 454.·8-s + (74.6 − 231. i)9-s − 97.0i·10-s − 337.·11-s + (−955. + 695. i)12-s − 682. i·13-s − 697.·14-s + (−85.8 − 117. i)15-s + 2.29e3·16-s + 513. i·17-s + ⋯
L(s)  = 1  − 1.83·2-s + (−0.808 + 0.588i)3-s + 2.36·4-s + 0.167i·5-s + (1.48 − 1.08i)6-s + 0.518·7-s − 2.51·8-s + (0.306 − 0.951i)9-s − 0.307i·10-s − 0.841·11-s + (−1.91 + 1.39i)12-s − 1.11i·13-s − 0.950·14-s + (−0.0984 − 0.135i)15-s + 2.24·16-s + 0.431i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.336i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.941 + 0.336i$
Analytic conductor: \(28.3879\)
Root analytic conductor: \(5.32803\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ -0.941 + 0.336i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.02279274396\)
\(L(\frac12)\) \(\approx\) \(0.02279274396\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (12.6 - 9.17i)T \)
59 \( 1 + (2.56e4 + 7.54e3i)T \)
good2 \( 1 + 10.3T + 32T^{2} \)
5 \( 1 - 9.35iT - 3.12e3T^{2} \)
7 \( 1 - 67.1T + 1.68e4T^{2} \)
11 \( 1 + 337.T + 1.61e5T^{2} \)
13 \( 1 + 682. iT - 3.71e5T^{2} \)
17 \( 1 - 513. iT - 1.41e6T^{2} \)
19 \( 1 - 189.T + 2.47e6T^{2} \)
23 \( 1 - 784.T + 6.43e6T^{2} \)
29 \( 1 - 4.83e3iT - 2.05e7T^{2} \)
31 \( 1 + 4.99e3iT - 2.86e7T^{2} \)
37 \( 1 - 9.26e3iT - 6.93e7T^{2} \)
41 \( 1 + 4.65e3iT - 1.15e8T^{2} \)
43 \( 1 - 5.28e3iT - 1.47e8T^{2} \)
47 \( 1 + 5.08e3T + 2.29e8T^{2} \)
53 \( 1 - 8.72e3iT - 4.18e8T^{2} \)
61 \( 1 - 2.66e4iT - 8.44e8T^{2} \)
67 \( 1 - 3.64e4iT - 1.35e9T^{2} \)
71 \( 1 + 8.35e4iT - 1.80e9T^{2} \)
73 \( 1 + 1.58e4iT - 2.07e9T^{2} \)
79 \( 1 + 2.00e4T + 3.07e9T^{2} \)
83 \( 1 - 6.94e4T + 3.93e9T^{2} \)
89 \( 1 + 2.13e4T + 5.58e9T^{2} \)
97 \( 1 - 1.51e5iT - 8.58e9T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.79614409806896592971221489867, −10.73651608300861565191800566380, −10.53933104824886174833668423510, −9.461340616229949435909506087475, −8.384908862735737768797915661217, −7.48910880146651507996641643946, −6.30501326415706515973057389883, −5.08902107977332946699712160522, −2.97919135533462185301286692197, −1.21712781070802172949687599879, 0.01776572594492171296601643852, 1.24133026726990805756373881732, 2.35048477536237204793226635131, 5.00457784685578759479125970491, 6.43349926839976692215331991562, 7.30932898981344696895584403798, 8.139205527482797328665480571879, 9.165197813626776974068160161616, 10.27649425677571960019033258531, 11.11785020299129855925395539960

Graph of the $Z$-function along the critical line